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Metamath Proof Explorer

Theorem oppfval

Description: Value of the opposite functor. (Contributed by Zhi Wang, 4-Nov-2025)

Ref Expression
Assertion oppfval ( 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 → ( 𝐹 oppFunc 𝐺 ) = ⟨ 𝐹 , tpos 𝐺 ⟩ )

Proof

Step Hyp Ref Expression
1 relfunc ⊢ Rel ( 𝐶 Func 𝐷 )
2 1 brrelex12i ⊢ ( 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 → ( 𝐹 ∈ V ∧ 𝐺 ∈ V ) )
3 oppfvalg ⊢ ( ( 𝐹 ∈ V ∧ 𝐺 ∈ V ) → ( 𝐹 oppFunc 𝐺 ) = if ( ( Rel 𝐺 ∧ Rel dom 𝐺 ) , ⟨ 𝐹 , tpos 𝐺 ⟩ , ∅ ) )
4 2 3 syl ⊢ ( 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 → ( 𝐹 oppFunc 𝐺 ) = if ( ( Rel 𝐺 ∧ Rel dom 𝐺 ) , ⟨ 𝐹 , tpos 𝐺 ⟩ , ∅ ) )
5 oppfvallem ⊢ ( 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 → ( Rel 𝐺 ∧ Rel dom 𝐺 ) )
6 5 iftrued ⊢ ( 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 → if ( ( Rel 𝐺 ∧ Rel dom 𝐺 ) , ⟨ 𝐹 , tpos 𝐺 ⟩ , ∅ ) = ⟨ 𝐹 , tpos 𝐺 ⟩ )
7 4 6 eqtrd ⊢ ( 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 → ( 𝐹 oppFunc 𝐺 ) = ⟨ 𝐹 , tpos 𝐺 ⟩ )